In this paper, two gradient properties of explicit convex functions are given.

Let pn(x) be a real algebraic polynomial of degree n, and consider the Lp norms on I = [−1, 1]. A classical result of A. A. Markoff states that if ‖pn‖. ∞ ≤ 1, then ‖P′n‖∞ ≤ n2. A generalization of Markoff's problem, first suggested by P. Turán, is to find upper bounds for ‖pn(J)‖p if ∣pn(x)∣≤ ψ(x)x ∈ I. Here ψ(x) is a given function, a curved majorant. In this paper we study extremal properties of ‖p′n‖2 and ‖p″n‖2 if pn(x) has the parabolic majorant ∣p(x)∣≤ 1 − x2, x ∈ I. We also consider the problem, motivated by a well-known result of S. Bernstein, of maximising ‖(1 − x2)

Some theorems on the existence of continuous real-valued functions on a topological space (for example, insertion, extension, and separation theorems) can be proved without involving uncountable unions of open sets. In particular, it is shown that well-known characterizations of normality (for example the Katětov-Tong insertion theorem, the Tietze extension theorem, Urysohn's lemma) are characterizations of normal σ-rings. Likewise, similar theorems about extremally disconnected spaces are true for σ-rings of a certain type. This σ-ring approach leads to general results on the existence of functions of class α.

In this paper we shall be concerned with the growth of functions in the class ℳp, where we write f ∈ ℳp 1 <p<∞, if:

(i) f is absolutely continuous on bounded subintervals of [0, ∞]);

(ii) f(0) = 0; and

(iii) .

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn , and independent copies A 1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

In this paper, the Opial's inequality, which has a wide range of applications in the study of differential and integral equations, is generalized to the case involving m functions of n variables, m, n ≥ 1.

Let X (1) ≦ X (2) ≦ ·· ·≦ X (N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt /t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.

The note re-examines Brown's new inequalities involving polynomials and fractional powers. Shorter proofs are provided, and greater attention is given to the conditions for the inequalities to hold.

Simple necessary and sufficient conditions for a function to be concave in terms of its shifted Laplace transform are given. As an application of this result, we show that the expected local time at zero of a reflected Lévy process with no negative jumps, starting from the origin, is a concave function of the time variable. A special case is the expected cumulative idle time in an M/G/1 queue. An immediate corollary is the concavity of the expected value of the reflected Lévy process itself. A special case is the virtual waiting time in an M/G/1 queue.

Two-sided inequalties for the ratio of modified Bessel functions of first kind are given, which provide sharper upper and lower bounds than had been known earlier. Wronskian type inequalities for Bessel functions are proved, and in the sequel alternative proofs of Turan-type inequalities for Bessel and modified Bessel functions are also discussed. These then lead to a two-sided inequality for Bessel functions. Also incorporated in the discussion is an inequality for the ratio of two Bessel functions for 0 < x < 1. Verifications of these inequalities are pointed out numerically.

We prove the higher integrability of nonnegative decreasing functions, verifying a reverse inequality, and we calculate the optimal integrability exponent for these functions.

This paper considers analogs of results on integral operators studied by Hörmander. Using the sharp function introduced by Fefferman and Stein, we prove weighted norm inequalities on kernel operators which map an Lp space into an Lq space, with q not equal to p. The techniques recover known results about fractional integral operators and apply to multiplier operators which satisfy a generalization of the Hörmander multiplier condition.

In this paper some new Opial-type integrodifferential inequalities in one variable are established. These generalize the existing ones which have a wide range of applications in the study of differential and integral equations.

The present paper deals with two inequalities which resemble Copson's integral inequalities. From our theorems, we obtain two interesting corollaries.

Various continuity conditions (in norm, in measure, weakly etc.) for the nonlinear superposition operator F x(s) = f(s, x(s)) between spaces of measurable functions are established in terms of the generating function f = f(s, u). In particular, a simple proof is given for the fact that, if F is continuous in measure, then f may be replaced by a function f which generates the same superposition operator F and satisfies the Carathéodory conditions. Moreover, it is shown that integral functional associated with the function f are proved.

This paper presents a new type of fractal surfaces called the Takagi surfaces. These are obtained by summing up pyramids of increasing (doubling) frequencies scaled by a geometric ratio b. The fractal dimension (box dimension) of the graph of these functions is shown to be log 8b/log 2.

Ramsey's theorem implies that every function f:0, 1ℝ isconvex or concave on an infinite set. We show that there is an upper semicontinuous function which is not convex or concave on any uncountable set. We investigate those functions which are not convex on any r element set (r). A typical result: if f is bounded from below and is not convex on any infiniteset then there exists an interval on which the graph of f can be covered by the graphs of countably many strictly concave functions.

A trigonometric series has “small gaps” if the difference of the orders of successive terms is bounded below by a number exceeding one. Wiener, Ingham and others have shown that if a function represented by such a series exhibits a certain behavior on a large enough subinterval I, this will have consequences for the behavior of the function on the whole circle group. Here we show that the assumption that f is in any one of various classes of functions of generalized bounded variation on I implies that the appropriate order condition holds for the magnitude of the Fourier coefficients. A generalized bounded variation condition coupled with a Zygmundtype condition on the modulus of continuity of the restriction of the function to I implies absolute convergence of the Fourier series.

We consider continuous Gaussian stochastic process indexed by a compact subset of a vector space over a local field. Under suitable conditions we obtain an asymptotic expression for the probability that such a process will exceed a high level. An important component in the proof of these results is a theorem of independent interest concerning the amount of ‘time’ which the process spends at high levels.

The concept of mixed invariant set is due to Bandt [1], Bedford [2], Dekking [3, 4], Marion [4] and Schulz [10]. An m-tuple B = (B1, …, Bm) of closed and bounded subsets Bi of a complete finitely compact (bounded and closed subsets are compact) metric space X is called a mixed invariant set with respect to contractions f1, …, fm and a transition matrix M = (mij), if, and only if,

for every i ∈ {1, …, m}. In the papers quoted an essential condition is that all mappings f1, …, fm be contractions. We will show that, under certain conditions, the construction of mixed invariant sets also works in cases where some of the mappings are isometries or even expanding mappings.